**Topology and Topological Space**

[box title=”Topics” style=”default” box_color=”#005ce6″ title_color=”#FFFFFF” radius=”3″]
- Introduction to Topology
- Applications of Topology
- What is Topology?
- What is a Euclidean space?
- Euclidean Axioms to Topological Axioms
- What is an Open Set & Open Interval?
- Who was Pavel Sergeyevich Alexandrov?
- Proof of Topology and Topological Space through Axioms

**Euclidean Axioms to Topological Axioms**

Topology is the abstract version of geometry. All geometrical shapes are topological spaces. We take some basic features of the geometrical spaces basic features and try to find out what will be the consequences of these features – axiom. Then only we can get a picture of geometrical results and their dependence on axioms.

Now, the mathematical proof is a watertight argument that begins with information, proceeds with a logical argument, and ends with proof. So to define a ‘topological space’ we need axioms. We will start with the axioms in Euclidean geometry

**Euclidean axioms:**

*If we take two points, rather any two points, a straight line can be drawn.**A line, which is terminated can be extended indefinitely*

**Topology axioms:**

*Let $X$ be a non-empty set. A set $T$ of subsets $X$ is said to be a topology on $X$ if:*

*(I) $X$ and the empty set $ϕ$ belongs to $T$*

*(II) $X$ the union of any finite or infinite number of sets $T$ belongs to $T$ and*

*(III) $X$ the intersection of any two sets $T$ belongs to $T$*

*The pair $(X,T)$ is called a topological space.*

Now that we have seen both the type of axioms, so we understand that from here our journey to define topology in mathematical terms would be to prove the axioms. Just as in Euclidean space we find axioms and proof, topology also follows the same rule.

Next | What is an Open Set & Open Interval? – Topology

See Also | More Topology Articles

**Further References:**

- Topology – James R. Munkres
- Topology without tears – Sidney A.Morris

**Article Advisor:**

- Richard Sot (PhD. Mathematics, University of Rochester, Rochester, NY)

**Website references:**

- https://mathworld.wolfram.com
- https://www.wikipedia.org/
- https://math.stackexchange.com/
- https://www.researchgate.net/publication/343635292

Shounak Bhattacharya is working as a Director, Training and Placement Department for Asian College of Teachers, Thailand. Apart from training, he is a researcher as well as a teacher in the area of

- Tensor analysis,
- General theory of relativity,
- Differential geometry and
- Introductory topology.

Shounak has been researching and educating the global crowd in the area of building career opportunities through teaching, researching, and communicating. Working with Asian College of Teachers, his main focus has been **increasing employability** factors among young students. The path leads to many factors including teaching, researching, communicating as well as creating a unique model which would employ the right person for the right job.

Relentlessly working in **simplifying difficult mathematical concepts**, Shounak’s interest revolves around Einstein’s general theory of relativity and creating structures, models, videos to educate the mass. He has been in touch with Prof. Lohiya, a former student of Dr. Stephen Hawking, and has conducted interviews regarding his experience working with Stephen Hawking.

He is working under the guidance of Dr.Richard Sot, Ph.D. Mathematics, University of Rochester, NY and Dr.Santosh Karn, Ph.D., Physics from Delhi University.

He has held numerous live interviews with researchers, educators, teachers, and dignitaries around the globe primarily,

- Thailand,
- Malaysia,
- Japan,
- The United Kingdom,
- Australia and
- Vietnam

He is a professional translator of the Arabic language, certified from RMIC. He is currently into Persian-Urdu translation of literary texts and creating videos on complex mathematical concepts.